Problem: Here's a parameterization of a cylinder, for $0 < \theta < 2\pi$ : $\vec{v}(\theta, z) = (r\cos(\theta), r\sin(\theta), z)$ What is the inward-pointing vector normal to the area element of this cylinder given $r = 2$, $\theta = \pi$, and $z = 3$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(-2, 0, 0)$ (Choice B) B $(2, 0, 2)$ (Choice C) C $(2, 0, 0)$ (Choice D) D $(2, 0, -2)$
Solution: The vector normal to the area element describes what's perpendicular to the surface, and the vector's magnitude represents the area of a tiny rectangle along the parameterization. After we compute it, we need to check whether it's pointing inwards or outwards. $\text{normal to area element} = \dfrac{\partial \vec{v}}{\partial \theta} \times \dfrac{\partial \vec{v}}{\partial z}$ Let's calculate the cross product. $\begin{aligned} \dfrac{\partial \vec{v}}{\partial \theta} \times \dfrac{\partial \vec{v}}{\partial z} &= \det \begin{pmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ \\ -r\sin(\theta) & r\cos(\theta) & 0 \\ \\ 0 & 0 & 1 \end{pmatrix} \\ \\ &= r\cos(\theta) \hat{\imath} + r\sin(\theta) \hat{\jmath} \end{aligned}$ Plugging in $r = 2$, $\theta = \pi$, and $z = 3$, we get the vector normal to the area element: $\begin{aligned} \dfrac{\partial \vec{v}}{\partial \theta} \times \dfrac{\partial \vec{v}}{\partial z} &= (-2, 0, 0) \end{aligned}$ The final step is to check whether this points inwards or outwards. The cylinder defined by $\vec{v}$ has an outward facing normal vector that points directly away from the origin, but with a zero $z$ -component. Because $\theta = \pi$ corresponds to a negative $x$, zero $y$, and zero $z$, we should have a positive $x$, zero $y$, and zero $z$. We need to reverse the signs of our calculation to make it match the inward normal vector. Therefore, the inward-pointing vector normal to the area element is $ (2, 0, 0)$.